We are given the equation:

\[
|4x - 8| + 1 = 5x
\]

### Step 1: Isolate the absolute value expression
First, subtract 1 from both sides to isolate the absolute value term:

\[
|4x - 8| = 5x - 1
\]

### Step 2: Break into two cases
The absolute value equation \( |A| = B \) can be written as two separate equations:

1. \( 4x - 8 = 5x - 1 \) (when \( 4x - 8 \geq 0 \))
2. \( 4x - 8 = -(5x - 1) \) (when \( 4x - 8 < 0 \))

### Case 1: \( 4x - 8 = 5x - 1 \)
Solve for \( x \):

\[
4x - 8 = 5x - 1
\]

Subtract \( 4x \) from both sides:

\[
-8 = x - 1
\]

Add 1 to both sides:

\[
-7 = x
\]

So, \( x = -7 \).

### Case 2: \( 4x - 8 = -(5x - 1) \)
Simplify the right-hand side:

\[
4x - 8 = -5x + 1
\]

Now solve for \( x \):

Add \( 5x \) to both sides:

\[
9x - 8 = 1
\]

Add 8 to both sides:

\[
9x = 9
\]

Divide by 9:

\[
x = 1
\]

### Step 3: Check for extraneous solutions
We need to check both solutions in the original equation because absolute value equations can sometimes introduce extraneous solutions.

- For \( x = -7 \):

\[
|4(-7) - 8| + 1 = 5(-7)
\]
\[
| -28 - 8 | + 1 = -35
\]
\[
| -36 | + 1 = -35
\]
\[
36 + 1 \neq -35
\]

Thus, \( x = -7 \) is not a valid solution.

- For \( x = 1 \):

\[
|4(1) - 8| + 1 = 5(1)
\]
\[
|4 - 8| + 1 = 5
\]
\[
|-4| + 1 = 5
\]
\[
4 + 1 = 5
\]

This is true, so \( x = 1 \) is the correct solution.

### Final Answer:
The solution to the equation is:

\[
x = 1
\]

Would you like more details or further clarification?

### Related Questions:
1. How would the solution change if the absolute value expression involved quadratic terms?
2. What happens if we modify the equation to \( |4x - 8| - 1 = 5x \)?
3. Can you explain the difference between an extraneous solution and a valid solution?
4. How do we solve absolute value inequalities, such as \( |4x - 8| \leq 5x \)?
5. What would the equation look like if the absolute value were outside the entire expression?

### Tip:
Always check potential solutions when solving absolute value equations to avoid including extraneous solutions.

Learn how to solve the absolute value equation |4x - 8| + 1 = 5x with a detailed step-by-step approach. Understand the process of isolating the absolute value, solving linear equations, and checking for extraneous solutions. Suitable for students in Grades 9-10.

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